## Stability of the minimal surface system and convexity of area functional

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- by Yng-Ing Lee and Mao-Pei Tsui PDF
- Trans. Amer. Math. Soc.
**366**(2014), 3357-3371 Request permission

## Abstract:

We study the convexity of the area functional for the graphs of maps with respect to the singular values of their differentials. Suppose that $f$ is a solution to the Dirichlet problem for the minimal surface system and the area functional is convex at $f$. Then the graph of $f$ is stable. New criteria for the stability of minimal graphs in any co-dimension are derived in the paper by this method. Our results in particular generalize the co-dimension one case, and improve the condition in the 2003 paper of the first author and M.-T. Wang from $|\bigwedge ^{2} df|\leq \frac {1}{p-1}$ to $|\bigwedge ^{2} df|\leq \frac {1}{\sqrt {p-1}},$ where $p$ is an upper bound of the rank of $df$, and the condition in the 2008 paper of the first author and M.-T. Wang from $\sqrt {det(I+(df)^Tdf)} \leq \frac {43}{40}$ to $\sqrt {det(I+(df)^Tdf)} \leq 2$.## References

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## Additional Information

**Yng-Ing Lee**- Affiliation: Department of Mathematics, National Taiwan University, Taipei, Taiwan — and — National Center for Theoretical Sciences, Taipei Office, Taipei, Taiwan
- Email: yilee@math.ntu.edu.tw
**Mao-Pei Tsui**- Affiliation: Department of Mathematics and Statistics, University of Toledo, Toledo, Ohio 43606
- MR Author ID: 278086
- Email: Mao-Pei.Tsui@Utoledo.edu
- Received by editor(s): November 23, 2011
- Published electronically: February 24, 2014
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**366**(2014), 3357-3371 - MSC (2010): Primary 53A10
- DOI: https://doi.org/10.1090/S0002-9947-2014-06223-5
- MathSciNet review: 3192598